2.2.11
Intuition Bada picture (YE KYU EXIST KARTA HAI)Ek 2D incompressible flow mein velocity field hota hai V = ( u , v ) do functions hain position ki. Lekin ye do functions independent nahi hain — inhe physics ek saath bandhti hai:
Incompressibility (mass conservation, ∇ ⋅ V = 0 ψ Irrotationality (koi spin nahi, ∇ × V = 0 ϕ
Toh do velocity components solve karne ki jagah, hum ek scalar field solve karte hain. Yahi poora point hai: kam unknowns, aasaan math.
Definition Velocity potentialAgar ek flow irrotational hai (∇ × V = 0 ϕ V = ∇ ϕ ⇒ u = ∂ x ∂ ϕ , v = ∂ y ∂ ϕ . potential flow kehte hain.
ϕ hamesha zero hota hai, ∇ × ( ∇ ϕ ) = 0 agar ∇ × V = 0 toh V ∇ ϕ
Intuition Electric potential se milta-julta kyun hai?Same math jaisi E = − ∇ V ϕ + badhte hue ϕ
Definition Stream function (2D incompressible)Agar ek flow incompressible (∇ ⋅ V = 0 ψ u = ∂ y ∂ ψ , v = − ∂ x ∂ ψ .
YE KYU KAAM KARTA HAI (DERIVE karo): 2D mein Continuity hai
∂ x ∂ u + ∂ y ∂ v = 0. u , v automatically satisfy ho. Try karo u = ∂ ψ / ∂ y v = − ∂ ψ / ∂ x ∂ x ∂ ( ∂ y ∂ ψ ) + ∂ y ∂ ( − ∂ x ∂ ψ ) = ψ x y − ψ y x = 0. Toh by construction, koi bhi ψ Isliye minus sign v
Intuition Streamlines = constantψ d x d y = u v u d y − v d x = 0 ( d x , d y ) ψ d ψ = ψ x d x + ψ y d y = − v d x + u d y . u d y − v d x d ψ = 0 streamlines constant ψ Kamaal hai.
Agar ek flow dono incompressible aur irrotational hai ("ideal flow"), toh ϕ ψ
Har ek Laplace's equation satisfy karta hai. KYU:
Incompressible + V = ∇ ϕ ∇ ⋅ ∇ ϕ = 0 ⇒ ∇ 2 ϕ = 0
Irrotational + ψ ∂ x ∂ v − ∂ y ∂ u = 0 ⇒ − ψ xx − ψ y y = 0 ⇒ ∇ 2 ψ = 0
∇ 2 ϕ = 0 , ∇ 2 ψ = 0
ϕ ψ ∇ ϕ = ( ϕ x , ϕ y ) = ( u , v ) ∇ ψ = ( ψ x , ψ y ) = ( − v , u ) ( u ) ( − v ) + ( v ) ( u ) = 0 Toh equipotential lines aur streamlines right angles par milti hain — ye ek orthogonal grid banati hain (ek "flow net").
Worked example Uniform flowLo ϕ = U x u = ϕ x = U v = ϕ y = 0 Ye step kyun? Potential ko differentiate karne se directly velocity milti hai.
Stream function: u = ψ y = U ⇒ ψ = U y v = − ψ x = 0 ψ = U y = Kyun? Constant y
Worked example Source flow (strengthm u r = 2 π r m u θ = 0 ϕ u r = ∂ ϕ / ∂ r ϕ = 2 π m ln r Kyun? Kyunki ∂ r ∂ 2 π m ln r = 2 π r m = u r u r = r 1 ∂ θ ∂ ψ = 2 π r m ⇒ ψ = 2 π m θ ψ = θ = Kyun? Ek source fluid ko seedha bahar dhakelta hai, toh streamlines spokes hoti hain.
Worked example Source ke liye orthogonality check karoEquipotentials ϕ = 2 π m ln r = r = 90° Ye kyun matter karta hai: flow-net theorem ko visually confirm karta hai.
Worked example Do streamlines ke beech fluxψ 1 ψ 2 Q = ψ 2 − ψ 1 θ = 0 θ = 2 π Q = 2 π m ( 2 π ) − 0 = m Kyun? Strength m m
ϕ Ye sahi kyun lagta hai: Velocity ek vector field hai, aur hum potentials pasand karte hain.
Fix: ϕ sirf tab jab flow irrotational ho (∇ × V = 0 ϕ
ψ Ye sahi kyun lagta hai: Streamlines hamesha exist karti hain, toh surely ψ Fix: Scalar ψ 2D incompressible flow ke liye guaranteed hai. Compressible flow ke liye tumhe ek modified (density-weighted) stream function chahiye.
Common mistake Minus sign bhool jaana:v = − ψ x Ye sahi kyun lagta hai: Symmetry tumhe v = ψ x Fix: Minus sign zaruri hai taaki continuity cancel ho (ψ x y − ψ y x = 0
Common mistake Roles mix karna: "ϕ Ye sahi kyun lagta hai: Dono ek scalar ki level curves hain.
Fix: ψ = streamlines (flow ke along). ϕ = equipotentials (flow ke perpendicular).
Recall Feynman: ek 12-saal ke bachche ko samjhaoEk nadi imagine karo. Paani left–right aur up–down directions mein alag-alag kitni tez chal rahi hai ye track karne ki jagah (do cheezein), hum har jagah ke liye EK clever number use karte hain:
Stream function ψ ψ Velocity potential ϕ ϕ
Mnemonic Definitions yaad karo"ψ ϕ
ψ u = ψ y v = − ψ x cross karte hain aur ek minus aata hai.ϕ u = ϕ x v = ϕ y match karte hain, koi minus nahi.
Aur "STeam = STreamlines" (ψ "Potential = Perpendicular Pressure-like" (ϕ
Velocity potential ϕ Flow irrotational honi chahiye,
∇ × V = 0 (kyunki
∇ × ∇ ϕ = 0 ).
Stream function ψ 2D incompressible flow,
∇ ⋅ V = 0 .
ϕ u , v u = ∂ ϕ / ∂ x , v = ∂ ϕ / ∂ y
ψ u , v u = ∂ ψ / ∂ y , v = − ∂ ψ / ∂ x
v = − ψ x Taaki continuity ψ x y − ψ y x = 0
ψ = Ek streamline (velocity ke tangent curve).
ϕ = Ek equipotential line, streamlines ke perpendicular.
Do streamlines ke beech flow rate Q = ψ 2 − ψ 1
Ideal flow mein ϕ ψ Laplace's equation, ∇ 2 ϕ = 0 , ∇ 2 ψ = 0
Equipotentials ⟂ streamlines kyun hain? ∇ ϕ ⋅ ∇ ψ = ( u ) ( − v ) + ( v ) ( u ) = 0
Cauchy–Riemann (fluid) relations ϕ x = ψ y ϕ y = − ψ x
Uniform flow U ϕ ψ ϕ = U x , ψ = U y
Strength m ϕ ψ ϕ = 2 π m ln r , ψ = 2 π m θ
Continuity Equation — stream function ka source (∇ ⋅ V = 0 Vorticity and Irrotational Flow — velocity potential ka source (∇ × V = 0 Laplace Equation — ϕ ψ Bernoulli Equation — tab use hoti hai jab ϕ Cauchy-Riemann Equations — complex potential w = ϕ + i ψ Electrostatic Potential — direct mathematical analogue (E = − ∇ V Flow Nets — ϕ ψ
auto-satisfies continuity
combined with incompressible
combined with irrotational
Incompressibility div V=0
